Some remarks on possible generalized inverses in semigroups

Jovan D. Kecki\'c

Abstract: For a given element $a$ of a semigroup $S$ it is possible that the system of equations in $x$: $axa = a$, $ax=xa$ is inconsistent, and that one or both systems $(S_k)$: $a^{k+1}x=a^k$, $ax=xa$ and $(\Sigma_k)$: $ axa=a$, $a^kx=xa^k $ are consistent for some positive integer $k$, in which case they have more than one solution. Some relations between those two systems are established. However, the chief aim of this note is to investigate the possibilities of extending $(S_k)$, by adding new balanced equations, so that this new system has unique solution. It is proved that if the extended system has unique solution, then the generalized inverse of $a$, defined by it, must be the Drazin inverse. It is also shown that the system $(\Sigma_2)$ $\wedge ax^2=x^2a\wedge xax=x$ cannot be extended into a system with unique solution.

Classification (MSC2000): 20M05

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